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Mirrors > Home > ILE Home > Th. List > tgidm | Unicode version |
Description: The topology generator function is idempotent. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tgidm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgvalex 12146 | . . . . 5 | |
2 | eltg3 12153 | . . . . 5 | |
3 | 1, 2 | syl 14 | . . . 4 |
4 | uniiun 3836 | . . . . . . . . . 10 | |
5 | simpr 109 | . . . . . . . . . . . . 13 | |
6 | 5 | sselda 3067 | . . . . . . . . . . . 12 |
7 | eltg4i 12151 | . . . . . . . . . . . 12 | |
8 | 6, 7 | syl 14 | . . . . . . . . . . 11 |
9 | 8 | iuneq2dv 3804 | . . . . . . . . . 10 |
10 | 4, 9 | syl5eq 2162 | . . . . . . . . 9 |
11 | iuncom4 3790 | . . . . . . . . 9 | |
12 | 10, 11 | syl6eq 2166 | . . . . . . . 8 |
13 | inss1 3266 | . . . . . . . . . . . 12 | |
14 | 13 | rgenw 2464 | . . . . . . . . . . 11 |
15 | iunss 3824 | . . . . . . . . . . 11 | |
16 | 14, 15 | mpbir 145 | . . . . . . . . . 10 |
17 | 16 | a1i 9 | . . . . . . . . 9 |
18 | eltg3i 12152 | . . . . . . . . 9 | |
19 | 17, 18 | sylan2 284 | . . . . . . . 8 |
20 | 12, 19 | eqeltrd 2194 | . . . . . . 7 |
21 | eleq1 2180 | . . . . . . 7 | |
22 | 20, 21 | syl5ibrcom 156 | . . . . . 6 |
23 | 22 | expimpd 360 | . . . . 5 |
24 | 23 | exlimdv 1775 | . . . 4 |
25 | 3, 24 | sylbid 149 | . . 3 |
26 | 25 | ssrdv 3073 | . 2 |
27 | bastg 12157 | . . 3 | |
28 | tgss 12159 | . . 3 | |
29 | 1, 27, 28 | syl2anc 408 | . 2 |
30 | 26, 29 | eqssd 3084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wex 1453 wcel 1465 wral 2393 cvv 2660 cin 3040 wss 3041 cpw 3480 cuni 3706 ciun 3783 cfv 5093 ctg 12062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-topgen 12068 |
This theorem is referenced by: tgss3 12174 txbasval 12363 |
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